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Primes p such that (p-1)^2+1 divides 2^(p-1)-1.
2

%I #22 Sep 08 2022 08:46:13

%S 17,257,8209,65537,649801

%N Primes p such that (p-1)^2+1 divides 2^(p-1)-1.

%C a(6), if it exists, is larger than 1.7*10^12. - _Giovanni Resta_, Jul 23 2015

%C N = 1382401 is the smallest composite number such that (n-1)^2+1 divides 2^(n-1)-1, cf. A260407; see also A081762 and A260406. The sequence contains all Fermat primes 2^2^k+1 > 5 (A019434). - _M. F. Hasler_, Jul 24 2015

%e 17 is in this sequence because (17 - 1)^2 + 1 = 257 divides 2^(17 - 1) - 1 = 65535; 65535 / 257 = 255.

%t fQ[n_] := PowerMod[2, n-1, (n-1)^2 + 1] == 1; p = 2; lst = {}; While[p < 10^9, If[ fQ@ p, AppendTo[lst, p]]; p = NextPrime@ p] (* _Robert G. Wilson v_, Jul 24 2015 *)

%o (Magma) [n: n in [1..2000000] | IsPrime(n) and (2^(n-1)-1) mod ((n-1)^2 + 1) eq 0]

%Y Cf. A081762 (primes p such that (p-1)^2 - 1 divides 2^(p-1) - 1).

%K nonn,more

%O 1,1

%A _Jaroslav Krizek_, Jul 22 2015